# Supervised learning with probabilistic morphisms and kernel mean embeddings

arXiv: https://arxiv.org/abs/2305.06348 · HTML: https://mlsanity.com/paper/2305.06348/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 66 references that are in the map.
- First posted on arXiv 10 May 2023.

## Builds on (most cited first, 7 of 7)

- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Sobolev Norm Learning Rates for Regularized Least-Squares Algorithm](https://mlsanity.com/paper/1702.07254/) (2017)
- [A Measure-Theoretic Approach to Kernel Conditional Mean Embeddings](https://mlsanity.com/paper/2002.03689/) (2020)
- [Disintegration and Bayesian Inversion via String Diagrams](https://mlsanity.com/paper/1709.00322/) (2017)
- [Representable Markov Categories and Comparison of Statistical Experiments in Categorical Probability](https://mlsanity.com/paper/2010.07416/) (2020)
- [Sobolev Norm Learning Rates for Conditional Mean Embeddings](https://mlsanity.com/paper/2105.07446/) (2021)
- [Probability, valuations, hyperspace: Three monads on Top and the support as a morphism](https://mlsanity.com/paper/1910.03752/) (2019)

## Built on by (most cited first, 1 of 1)

- [Categorical and geometric methods in statistical, manifold, and machine learning](https://mlsanity.com/paper/2505.03862/) (2025)

## Abstract (arXiv)

In this paper I propose a generative model of supervised learning that unifies two approaches to supervised learning, using a concept of a correct loss function. Addressing two measurability problems, which have been ignored in statistical learning theory, I propose to use convergence in outer probability to characterize the consistency of a learning algorithm. Building upon these results, I extend a result due to Cucker-Smale, which addresses the learnability of a regression model, to the setting of a conditional probability estimation problem. Additionally, I present a variant of Vapnik-Stefanuyk's regularization method for solving stochastic ill-posed problems, and using it to prove the generalizability of overparameterized supervised learning models.